Defect Engineering of Disordered Carbon Anodes with Ultra-High Heteroatom Doping Through a Supermolecule-Mediated Strategy for Potassium-Ion Hybrid Capacitors

Highlights The N/S co-doped lignin-derived porous carbon (NSLPCs) with ultra-high heteroatom doping was prepared through a novel supramolecule-mediated pyrolysis strategy. Covalently bonded graphitic carbon/amorphous carbon intermediates induce the formation of high heteroatom doping. The high heteroatom doping of NSLPC could provide abundant defective active sites for the adsorption of K+. Supplementary Information The online version contains supplementary material available at 10.1007/s40820-022-01006-0.


S1.1 Electrochemical Equations and Calculations
In PIHCs full-cell tests, the calculations of energy (E, Wh kg -1 ) and power densities (P, W kg -1 ) based on the total mass of both anode and cathode materials were performed using the equations below [S1, S2]: = ∆ × (S1) = × 3600 ⁄ (S2) ∆ = ( + )/2 (S3) in which t (s) is the discharge time, i (A g -1 ) is the charge/discharge current, Vmax (V) is the discharge potential excluding the IR drop, and Vmin (V) is the potential at the end of discharge voltages.
The contributions to capacitive controlled and diffusion-controlled processes are estimated as follows [S3]: = (S4) S2/S19 where is the instantaneous current density, v is the scanning rate, and a and b are adjustable parameters obtained from the v1 and v2 scanning rates, respectively. The electrode reaction is controlled by the diffusion process when the value of b is 0.5, and the electrode response is controlled by the surface-dominated process when the b value is 1. The contributions of the capacitive current and the diffusion-controlled current to the whole electrode reaction process are represented by the k1v and k2v 1/2 in equations (S5) and (S6), respectively. The diffusion coefficients of K + were calculated from the Galvanostatic Intermittent Titration Technique (GITT) using the following equation [S4].
Where τ denotes the duration of the current pulse, s; mB is the mass of electrode active material, g; S is the geometric area of the electrode, cm 2 ; ΔEs is the quasi-thermodynamic equilibrium potential difference before and after the current pulse, V; ΔEt is the potential difference during current pulse neglecting the IR-drop, V; VM is the molar volume of the active materials, MB is the molar mass of carbon. The value of MB/VM is the density of electrodes (1.5 g cm -3 ), which is estimated to be lower than the density of graphite (2.2 g cm -3 ). Before the GITT measurement, the NSLPCs electrodes were pre-cycled for three cycles under a constant current density of 50 mA g -1 .

S2 Molecular Dynamic Simulation and Density Functional Theory Calculation
Firstly, Hundreds of initial configurations were generated by the genmer module of the Molclus program [S5]. All the clusters were then pre-optimized under the semi-empirical method GFN2-Xtb [S6]. The obtained configurations with the low energy were further optimized and calculated the frequency under B3LYP-D3(BJ)/6-31G* level using Gaussian16 package with the consideration of implicit solvent model. The interaction energy between the LS and MA+TCA and the energy among the MA and TCA molecules were calculated with the correction of basis set superposition error (BSSE) under the B3LYP-D3(BJ)/6-311G** level.
To study the weak interaction among the molecules, the Independent gradient model based on Hirshfeld partition (IGMH) [S7] were analyzed by Multiwfn 3.8 (dev) program [S8] and visualized by VMD 1.9.4. software [S9]. All the calculations are performed in the framework of the density functional theory with the projector augmented plane-wave method, as implemented in the Vienna ab initio simulation package [S10]. The generalized gradient approximation proposed by Perdew, Burke, and Ernzerhof is selected for the exchange-correlation potential [S11]. The long-range van der Waals interaction is described by the DFT-D3 approach [S12]. The cut-off energy for the plane wave is set to 500 eV. The energy criterion is set to 10-6 eV in the iterative solution of the Kohn-Sham equation. The Brillouin zone integration is performed at the Gamma point with a 2 × 2 × 1 k-mesh grid for structural optimization calculations and a 6×6×1 k-mesh grid for the electronic structure calculations. All the structures are relaxed until the residual forces on the atoms have declined to less than 0.03 eV Å -1 . The vacuum thickness along the z-axis is set to 15 Å, which is large enough to avoid interaction between periodic images.       The cycling performance of PIHCs at a current density of 0.2 A g -1 (The specific capacity was calculated based on the total mass of the anode and cathode electrodes)